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A Cow, A Monkey and a Tree

Q: A Cow is tethered to a tree by a rope that has half the length of the circumference of the tree's stump. A monkey with a stone in hand jumps onto the tree and starts hopping around the tree's branches which covers a circular area of radius five times that of the stump. This causes the cow to run about at random and monkey drops the stone at random. What is the probability that the cow would get hit?

A: The situation can be characterized by the figure shown below (not to scale)

The monkey can be anywhere in the light shaded orange area before it drops the stone (this of course assumes that the stump has no terrace area). The cow, can sweep a spiral as shown in the figure below along either lines (poorly drawn).

To start, lets first compute the area the cow can sweep. Assume the cow moves a bit to the left, the rope would wind up along the stump of the tree. To further formalize this assume it sweeps out an angle \(\theta\) at the center as shown figure below.

In the figure if the cow's rope winds up by a length AB subtending an angle \(\theta\) at the center C, and if the cow moves from D to E an incremental angle of \(d\theta \) is subtended at the centre. The length of the rope is now decreased from \(\pi r\) to \(\pi r - r\theta\). The length of DE can be computed as
$$
(\pi r - r\theta )d\theta
$$
The incremental area of triangle ADE is \(\frac{1}{2}\times DE \times AD\) which works out as
$$
dA = \frac{1}{2}(\pi r - r \theta)^{2}d\theta
$$
To compute the total area, we can integrate out with \(\theta\) running from \([0,\pi]\) and as the area is symmetric, we multiply by 2
$$
A =2 \times \int_{0}^{\pi}\frac{1}{2}r^{2}(\pi - \theta)^{2}d\theta
$$
The above integral simplifies to
$$
A = \frac{r^{2}\pi^{3}}{3}
$$
The total area the monkey can drop the stone is easy to compute as
$$
A_{total} = \pi (5r)^{2} - \pi r^{2} = 24\pi r^{2}
$$
The sought area and probability is simply the ratio of the two areas \(\frac{A}{A_{total}} = \frac{\pi^{3}}{24\times 3} \approx 43\%\)

Discovering Statistics Using R
This is a good book if you are new to statistics & probability while simultaneously getting started with a programming language. The book supports R and is written in a casual humorous way making it an easy read. Great for beginners. Some of the data on the companion website could be missing.

A Course in Probability Theory, Third Edition
Covered in this book are the central limit theorem and other graduate topics in probability. You will need to brush up on some mathematics before you dive in but most of that can be done online

Discovering Statistics Using R
This is a good book if you are new to statistics & probability while simultaneously getting started with a programming language. The book supports R and is written in a casual humorous way making it an easy read. Great for beginners. Some of the data on the companion website could be missing.

Linear Algebra (Dover Books on Mathematics)
An excellent book to own if you are looking to get into, or want to understand linear algebra. Please keep in mind that you need to have some basic mathematical background before you can use this book.

Linear Algebra Done Right (Undergraduate Texts in Mathematics)
A great book that exposes the method of proof as it used in Linear Algebra. This book is not for the beginner though. You do need some prior knowledge of the basics at least. It would be a good add-on to an existing course you are doing in Linear Algebra.

Follow @ProbabilityPuzIf you are looking to learn time series analysis, the following are some of the best books in time series analysis.

Introductory Time Series with R (Use R!)
This is good book to get one started on time series. A nice aspect of this book is that it has examples in R and some of the data is part of standard R packages which makes good introductory material for learning the R language too. That said this is not exactly a graduate level book, and some of the data links in the book may not be valid.

Econometrics
A great book if you are in an economics stream or want to get into it. The nice thing in the book is it tries to bring out a oneness in all the methods used. Econ majors need to be up-to speed on the grounding mathematics for time series analysis to use this book. Outside of those prerequisites, this is one of the best books on econometrics and time series analysis.